Continuous Bivariate Distributions Research Articles

Fisher Information and Shannon’s Entropy for Record Values and Their Concomitants under Iterated FGM Family

Let {(Xi ,Yi), i ≥ 1} be independent and identically distributed random variables (RVs) from a continuous bivariate distribution. If {Rn,n ≥ 1} is the sequence of upper record values in the sequence {Xi}, then the RV Yi, which corresponds to Rn is called the concomitant of the nth record, denoted by R[n]. We study the Shannon entropy (SHANE) of R[n] and (Rn,R[n]) under iterated Farlie-Gumbel-Morgenstern (IFGM) family. In addition, we find the Kullback-Leibler distance (K-L) between R[n] and Rn. Moreover, we study the Fisher information matrix (FIM) for record values and their concomitants about the shape-parameter vector of the IFGM family. Also, we study the relative efficiency matrix of that vector-estimator of the shape-parameter vector whose covariance matrix is equal to Cramer-Rao lower bound, based on record ´ values and their concomitants. In addition, the Fisher information number (FIN) of R[n] is derived. Finally, we evaluate the FI about the mean of exponential distribution in the concomitants of record values.

New continuous bivariate distributions generated from shock models

In this paper, we develop a new class of continuous bivariate distributions based on a shock model. The proposed class is general in the sense that numerous families of distributions can be generated from it. Distributional properties of the proposed class of distributions are studied. The bivariate ageing property and the dependence structure are also analysed. It is shown that the developed bivariate distributions exhibit positive dependency and a well-defined subclass of the proposed class of bivariate distributions satisfy the bivariate lack of memory property. Specific families of distributions are derived and applied to a real data set.

An EM algorithm for absolutely continuous Marshall-Olkin bivariate Pareto distribution with location and scale

Block and Basu bivariate exponential distribution is one of the most popular absolutely continuous bivariate distributions. In this article, we have considered a Block–Basu type bivariate Pareto distribution. Here in the standard manner, first Marshall-Olkin type singular bivariate distribution has been constructed, and then by taking away the singular component similar to the Block and Basu model, an absolute continuous BB-BVPA model has been constructed. Further, the location and scale parameters also have been introduced. Therefore, the model has seven parameters. We drive different properties of this absolutely continuous distribution. Since the maximum likelihood estimators of the parameters cannot be expressed in a closed form, we propose to use an EM algorithm to compute the estimators of the model parameters. Some simulation experiments have been performed for illustrative purposes. We fit the model to rainfall data in the context of landslide risk estimation. The methods are implemented in the publicly available R package bvpa.

A first-order Stein characterization for absolutely continuous bivariate distributions

A random variable X has a standard normal distribution if and only if E [ f ′ ( X ) ] = E [ X f ( X ) ] for any continuous and piecewise continuously differentiable function f such that the expectations exist. This first-order characterizing equation, called the Stein identity, has been extended to other univariate distributions. For the multivariate normal distribution, a number of Stein identities have already been developed, all of them second order equations. In this study, we developed a new Stein characterization for the bivariate normal distribution. Unlike many existing multivariate versions in the literature, ours is a system of first-order equations which has the univariate Stein identity as a special case. We also constructed a generalized Stein characterization for other absolutely continuous bivariate distributions. Finally, we illustrated how this Stein characterization looks like for some known absolutely continuous bivariate distributions.

Bivariate Continuous Negatively Correlated Proportional Models with Applications in Schizophrenia Research

Bivariate continuous negatively correlated proportional data defined in the unit square (0,1)2 often appear in many different disciplines, such as medical studies, clinical trials and so on. To model this type of data, the paper proposes two new bivariate continuous distributions (i.e., negatively correlated proportional inverse Gaussian (NPIG) and negatively correlated proportional gamma (NPGA) distributions) for the first time and provides corresponding distributional properties. Two mean regression models are further developed for data with covariates. The normalized expectation–maximization (N-EM) algorithm and the gradient descent algorithm are combined to obtain the maximum likelihood estimates of parameters of interest. Simulations studies are conducted, and a data set of cortical thickness for schizophrenia is used to illustrate the proposed methods. According to our analysis between patients and controls of cortical thickness in typical mutual inhibitory brain regions, we verified the compensatory of cortical thickness in patients with schizophrenia and found its negative correlation with age.

A New Approach to Bivariate Transmutation: Construction of Continuous Bivariate Distribution Under Negative Dependency

In this study, a new approach to transmutation theory is developed by using negative dependency basement. Once choosing a distribution that has negative dependency with the same marginal, a new bivariate distribution is derived. In this study, we examined a new transmutation technique in which a negative dependency offers a big success in modeling rather than most known and used statistical distributions. This approach clash with classical transmutation methods. In this study at the beginning, the classical transmutation is defined. Later, we introduce the new technique and obtain lower and upper bounds of distribution to show that this approach gives us a distribution. Gaining new bivariate continuous distributions with this technique may be more appropriate in theory, and modeling of some data sets in terms of this approach may be more efficient.

APPLICATIONS OF MULTIVARIATE AND BIVARIATE ANALYSIS IN SCIENCE AND ENGINEERING

The applicability of multivariate analysis provides the gradual build up for the discussion of multivariate problems, it extends the application procedure to bi-variety distributions. Sample from a multivariate population looks like and how the sample statistics like mean, variance, covariance matrix, correlation matrix can be calculated are explained with the help of a numerical practical illustrations. It has wide applications in many fields like education, medicines, agriculture and Economics. A series of univariate statistical analysis carried out separately for each of the variables. The objectives of scientific investigations for which multivariate methods most naturally implemented in statistical computation methods. Whenever an investigator requires to understand a social or physical phenomena variables to records. Bivariate discrete and continuous random vectors are considered in expansion form of Multifivatirate analysis tabulations. Bivariate distribution which has got an importance of its own will be explained for easy understanding of various methods of analysis in Multivariate and Bivariate application methods. It provides the methods of multivariate techniques, data organization, arrays and basic descriptive statistics, Bivariate continuous distributions methods, expectation of random variables, and marginal mass functions role in Engineering curriculum.

A partial likelihood-based two-dimensional multistate markov model with application to myocardial infarction and stroke recurrence

Myocardial infarction (MI) and stroke are the most common acute life-threatening cardiovascular disease (CVD) events and are triggered by the rupture of lipid-rich plaques in the arterial wall during the progression of atherosclerosis. Thus, they share a common pathology and preventive interventions. However, the association between the recurrent events of MI and stroke among CVD patients as the disease progresses remains unclear. In this study, we propose a bivariate Markov model to describe the multistate of recurrent events of MI and stroke. A partial likelihood approach was adopted by using the Sarkar's absolutely continuous bivariate exponential distribution (ACBVE) separately for the transitions among different states. The parametric model estimates the hazard function at each state and thus takes more information than an alternative semiparametric approach. As an illustrative example, we analyzed recurrent events of MI and stroke in individuals from the Taiwan National Health Insurance Research Database. Comparisons with the nonparametric Aalen-Johansen estimator for each state showed that the parametric ACBVE explained the data well. The correlation coefficients between the first recurrent MI and stroke tended to increase as the state of disease status progresses. The proposed two-dimensional multistate Markov model may be employed to describe the progressive comorbidity of two associated diseases.

Construction of Continuous Bivariate Distribution by Transmuting Dependent Distribution

In this study, a new bivariate distribution family is introduced by adding an appropriate term to independent class. By choosing a base distribution which is negatively dependent from the same marginals we derive a new distribution around the product of marginals, i.e. independent class of distribution. We note that the new distribution has additional parameter which would provide additional flexibility in applications. The joint probability density, joint reliability and reversed hazard rate functions of the new bivariate distribution are obtained. Furthermore, we obtain lower and upper bounds of Spearman’s correlation coefficient. Two example are given to illustrate this family. This new bivariate continuous distribution can make more appropriate modeling of some data sets in terms of the Spearman rank coefficient.

Correction to: Discrete analogues of continuous bivariate probability distributions

This erratum is published as vendor overlooked corrections on Table 6 and deleted two references.

Discrete analogues of continuous bivariate probability distributions

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.

Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data

We discuss the testing problem of homogeneity of the marginal distributions of a continuous bivariate distribution based on a paired sample with possibly missing components (missing completely at random). Applying the well-known two-sample Cramer–von-Mises distance to the remaining data, we determine the limiting null distribution of our test statistic in this situation. It is seen that a new resampling approach is appropriate for the approximation of the unknown null distribution. We prove that the resulting test asymptotically reaches the significance level and is consistent. Properties of the test under local alternatives are pointed out as well. Simulations investigate the quality of the approximation and the power of the new approach in the finite sample case. As an illustration we apply the test to real data sets.

On the asymptotic covariance of the multivariate empirical copula process

Abstract Genest and Segers (2010) gave conditions under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance than the standard empirical process based on a random sample from the underlying copula. An extension of this result to the multivariate case is provided.

On characterizations of some bivariate continuous distributions by properties of higher-degree bivariate stop-loss transforms

In this paper we study some characteristic properties of higher-degree bivariate stop-loss transforms (partial moments). A new bivariate distribution is proposed by extending the characterizing identity of univariate partial moments due to Lin (2003) to the bivariate case. A real-data analysis is also carried out to illustrate the theoretical results.

Multi-Panel Kendall plot in light of an ROC curve analysis applied to measuring dependence

ABSTRACTThe Kendall plot (-plot) is a plot measuring dependence between the components of a bivariate random variable. The -plot graphs the Kendall distribution function against the distribution function of VU, where V and U are independent uniform random variables. We associate -plots with the receiver operating characteristic () curve, a well-accepted graphical tool in biostatistics for evaluating the ability of a biomarker to discriminate between two populations. The most commonly used global index of diagnostic accuracy of biomarkers is the area under the curve (). In parallel with the , we propose a novel strategy to measure association between random variables from a continuous bivariate distribution. First, we discuss why the area under the conventional Kendall curve () cannot be used as an index of dependence. We then suggest a simple and meaningful extension of the definition of the -plots, and define an index of dependence that is based on . This measure characterizes a wide range of two-variable relationships, thereby completely detecting the underlying dependence structure. Properties of the proposed index satisfy the mathematical definition of a measure. Finally, simulations and real data examples illustrate the applicability of the proposed method.

Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach

ABSTRACTThe modeling and analysis of lifetime data in which the main endpoints are the times when an event of interest occurs is of great interest in medical studies. In these studies, it is common that two or more lifetimes associated with the same unit such as the times to deterioration levels or the times to reaction to a treatment in pairs of organs like lungs, kidneys, eyes or ears. In medical applications, it is also possible that a cure rate is present and needed to be modeled with lifetime data with long-term survivors. This paper presented a comparative study under a Bayesian approach among some existing continuous and discrete bivariate distributions such as the bivariate exponential distributions and the bivariate geometric distributions in presence of cure rate, censored data and covariates. In presence of lifetimes related to cured patients, it is assumed standard mixture cure rate models in the data analysis. The posterior summaries of interest are obtained using Markov Chain Monte Carlo methods. To illustrate the proposed methodology two real medical data sets are considered.

Basu-Dhar's bivariate geometric distribution in presence of censored data and covariates: some computational aspects

Some computational aspects to obtain classical and Bayesian inferences for the Basu and Dhar (1995) bivariate geometric distribution in presence of censored data and covariates are discussed in this paper. The posterior summaries of interest are obtained using standard existing MCMC (Markov Chain Monte Carlo) simulation methods available in popular free softwares as the OpenBugs software and the R software. Numerical illustrations are introduced considering simulated and real datasets showing that the use of discrete bivariate distributions may be a good alternative to the use of continuous bivariate distributions, in many areas of application.

First-order dominance: stronger characterization and a bivariate checking algorithm

How to determine whether one distribution first-order dominates another is a fundamental problem that has many applications in economics, finance, probability theory, and statistics. Nevertheless, little is known about how to efficiently check first-order dominance for finite multivariate distributions. Utilizing that this problem can be formulated as a transportation problem with a special structure, we provide a stronger characterization of multivariate first-order dominance and develop a linear time complexity checking algorithm for the bivariate case. We illustrate the use of the checking algorithm when numerically assessing first-order dominance among continuous bivariate distributions.

Dependence structure and test of independence for some well-known bivariate distributions

In this paper, we study the dependence structure of some bivariate distribution functions based on dependence measures of Kochar and Gupta (Biometrika 74(3):664–666, 1987) and Shetty and Pandit (Stat Methods Appl 12:5–17, 2003) and then compare these measures with Spearman’s rho and Kendall’s tau. Moreover, the empirical power of the class of distribution-free tests introduced by Kochar and Gupta (1987) and Shetty and Pandit (2003) is computed based on exact and asymptotic distribution of U-statistics. Our results are obtained from simulation work in some continuous bivariate distributions for the sample of sizes \(n=6,8,15,20\) and 50. Also, we apply some examples to illustrate the results. Finally, we compare the common estimators of dependence parameter based on empirical MSE.

A class of continuous bivariate distributions with linear sum of hazard gradient components

The main purpose of this article is to characterize a class of bivariate continuous non-negative distributions such that the sum of the components of underlying hazard gradient vector is a linear function of its arguments. It happens that this class is a stronger version of the Sibuya-type bivariate lack of memory property. Such a class is allowed to have only certain marginal distributions and the corresponding restrictions are given in terms of marginal densities and hazard rates. We illustrate the methodology developed by examples, obtaining two extended versions of the bivariate Gumbel’s law.